So it still fits the definition of perfect secrecy, but (with vanishingly small probability when the key is large) it's possible for a key to be chosen that is more "guessable"?

I don't think this is a problem. Depending on the guess for the key an attacker could get back any possible message as the plaintext. So there is no reason for them to believe that the message they obtain by using a simple key (e.g. AAAAAAAAAAAA as in the example) is the original plaintext?